Tensor network implementation of bulk entanglement
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Hsieh, Timothy H., Liang Fu, and Xiao-Liang Qi. “Tensor
Network Implementation of Bulk Entanglement Spectrum.” Phys.
Rev. B 90, no. 8 (August 2014). © 2014 American Physical
Society
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http://dx.doi.org/10.1103/PhysRevB.90.085137
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PHYSICAL REVIEW B 90, 085137 (2014)
Tensor network implementation of bulk entanglement spectrum
Timothy H. Hsieh,1 Liang Fu,1 and Xiao-Liang Qi2
1
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
2
Department of Physics, Stanford University, Stanford, California 94305, USA
(Received 3 July 2014; published 25 August 2014)
Many topologically nontrivial states of matter possess gapless degrees of freedom on the boundary, and when
these boundary states delocalize into the bulk, a phase transition occurs, and the system becomes topologically
trivial. We show that tensor networks provide a natural framework for analyzing such topological phase transitions
in terms of the boundary degrees of freedom which mediate it. To do so, we make use of a correspondence between
a topologically nontrivial ground state and its phase transition to a trivial phase established in T. Hsieh and
L. Fu (arXiv:1305.1949). This involved computing the bulk entanglement spectrum (BES) of the ground state
upon tracing out an extensive subsystem. This work implements BES via tensor network representations of ground
states. In this framework, the universality class of the quantum critical entanglement Hamiltonian in d spatial
dimensions is either derived analytically or mapped to a classical statistical model in d + 1 dimensions, which can
be studied using Monte Carlo or tensor renormalization-group methods. As an example, we analytically derive
the universality classes of topological phase transitions from the spin-1 chain Haldane phase and demonstrate that
the Affleck-Kennedy-Lieb-Tasaki (AKLT) wave function (and its generalizations) remarkably contains critical
six-vertex (and, in general, eight-vertex) models within it.
DOI: 10.1103/PhysRevB.90.085137
PACS number(s): 03.67.Mn, 75.10.Pq, 03.65.Ud
I. INTRODUCTION
Despite lacking a local order parameter, topological states
contain a wealth of subtly encoded information [1–8], including in some cases topological invariants such as the
Chern number and in other cases ground-state degeneracy on
higher-genus manifolds, for example. Given the stark contrast
between topological states and classically ordered states, it
is natural that purely quantum notions are often necessary
for analyzing topological states. In particular, measures of
entanglement, which has no classical analog, have proven to
be extremely useful. For example, the entanglement entropy
between a subsystem and its complement has been used
[9–13] to detect topological order in a ground state. Moreover,
the full spectrum of the reduced density matrix, called the
entanglement spectrum [14–22], has allowed simulation of the
edge excitations of a topological ground state. In other words,
tracing out a subsystem from a ground state achieves an effect
similar to introducing excitations localized at the boundary of
the subsystem.
With this paradigm in mind, two of us have introduced
a technique called the bulk entanglement spectrum (BES) to
study the bulk of a system [23]. Specifically, it was found
that a topological state contains information about its phase
transition to a trivial phase. How is it possible that a single wave
function can give birth to quantum criticality associated with a
topological phase transition? By using a special partition of the
topological state that is extensive in all directions and possesses
symmetry between the remaining and traced out subsystems,
the resulting bulk entanglement Hamiltonian [see Eq. (1)]
was argued either to be critical or to possess ground-state
degeneracy. In the former case, the critical bulk entanglement
Hamiltonian sits right at a phase transition between the original
topologically nontrivial phase and a trivial phase. The essence
of this argument is the discrete nature of topological order
and extreme limits of the partition. When nearly nothing is
traced out, the remaining subsystem is in the same topological
1098-0121/2014/90(8)/085137(5)
phase as the original nontrivial wave function. When nearly
everything is traced out, the remaining subsystem consists of
decoupled small “islands” and thus is topologically trivial.
Tuning the geometry of the partition thus induces a phase
transition in the entanglement Hamiltonian. If there is a
single phase transition, then it must occur at the intermediate,
symmetric partition defined above. This protocol, which is
derived from a single wave function, differs markedly from
the usual realization of a quantum phase transition by tuning
parameters in a Hamiltonian. As a proof of concept, BES has
rigorously been shown to work for integer quantum Hall states.
The BES produces the massless Dirac spectrum expected at
the transition between states with Chern numbers 1 and 0.
In this paper, we study the quantum phase transition in
BES for generic matrix product states (MPSs) [24,25], which
are efficient representations of generic noncritical states in
one dimension. We analytically implement the BES technique
and obtain a more explicit understanding of the entanglement
Hamiltonian and topological phase transition. The partition
function of this bulk entanglement Hamiltonian serves as
the centerpiece of this work, with three primary uses. First,
it allows us to identify the critical theory of the entanglement Hamiltonian. Second, it maps the quantum critical
d-dimensional system to a classical (d + 1)-dimensional
system, enabling Monte Carlo numerics or tensor
renormalization-group methods [26] to tackle quantum critical problems. Finally, it provides a dynamic picture of a
topological phase transition and highlights the importance
of edge states in mediating such a phase transition. We
will explicitly show that the virtual degree of freedom in
the MPSs gains a life of its own in the partition function,
and it is precisely the interactions of these virtual elements
which constitute the topological phase transition. While we
focus on one-dimensional systems in this work, many of our
techniques generalize to tensor network wave functions in any
dimension.
085137-1
©2014 American Physical Society
TIMOTHY H. HSIEH, LIANG FU, AND XIAO-LIANG QI
A
B
β
δ
α
γ
PHYSICAL REVIEW B 90, 085137 (2014)
FIG. 1. (a) A segment of a matrix product state partitioned into
two spatial subspaces, A and B. The open-ended vertical links
represent physical degrees of freedom, and the horizontal links
represent “virtual” degrees of freedom which are summed as in
Eq. (2). (b) The reduced density matrix obtained by tracing out B.
The MPS transfer matrix is shown by the box.
II. BULK ENTANGLEMENT SPECTRUM AND TENSOR
NETWORKS
We begin by briefly reviewing BES and tensor networks
before using them together. Consider a ground state |
defined on a Hilbert space S and partition S into two
complementary parts, A and B. After tracing out part B, the
description of | on A is given by a thermal density matrix
ρA = TrB || ≡ e−HA ,
(1)
corresponding to an entanglement Hamiltonian HA . The
entanglement spectrum is the set of eigenvalues of HA , and
in the following we will be interested in the ground state of
HA and its topological nature.
When an extensive partition is used, i.e., when A and B
are extensive with system size in all directions, one attains
a bulk entanglement Hamiltonian: HA has support on an
extensive subsystem A. It was argued in [23] that when the
ground state | is an irreducible topological state [27] and
when A and B are related by symmetry, HA either (1) has
ground-state degeneracy or (2) is gapless and characterizes
a topological phase transition from nontrivial to trivial. As
outlined earlier, this claim can be motivated by considering
extreme examples of extensive partitions. The critical point
occurs at the symmetric partition in which A and B are related
by a symmetry such as translation.
Tracing out degrees of freedom has a convenient pictorial
representation in the framework of tensor networks. Consider a
σ
MPS given by a tensor M{α}
with a physical index σ and virtual
indices {α} emanating from the physical sites [see Fig. 1(a)].
The virtual indices are contracted, leaving a wave function |ψ
defined by
σ1 · · · σN |ψ ≡
M σ1 · · · M σN .
(2)
virtual indices
The reduced density matrix ρA obtained from tracing out part
B is
ρA =
σB |ψψ|σB .
(3)
σB
Graphically, ψ| is simply represented by reflecting the MPS
|ψ and complex conjugating the tensors. Then, the σB
indices are contracted to yield ρA [Fig. 1(b)]. In this pictorial
language, the topological phase transition realized by tuning
the extensive partition is shown in Fig. 2, in which the density
matrix interpolates between the nontrivial projector onto the
topological ground state and a trivial product of density
FIG. 2. The tensor network representation of the BES procedure.
The reduced density matrices ρA corresponding to different extensive
partitions are shown. Tuning the extensive partition of a topologically
nontrivial ground state realizes a topological phase transition that
occurs at the partition where A and B are symmetric (middle). If too
little is traced out (top), ρA ≈ || is nontrivial. If too much is
traced out (bottom), ρA ≈ ⊗i∈A ρi is trivial.
operators. The phase transition occurs at a partition in which
A and B are related by some symmetry.
III. PARTITION FUNCTION OF ENTANGLEMENT
HAMILTONIAN
It is now extremely useful to construct the partition
function of the entanglement Hamiltonian Z = tr(e−nHA ) =
tr(ρAn ), where n = 1/T is the inverse “temperature.” We
will eventually take the limit T → 0 (n → ∞) to probe the
universality class of HA . Graphically, one simply stacks n
copies of ρA and then contracts all physical indices, including
those at the top of the nth copy and those at the bottom of
the first copy. Because all indices are contracted, we now
have the freedom to reinterpret the partition function as one
involving the virtual degrees of freedom, thus providing a
different perspective on the topological phase transition being
studied. In some cases, we can rewrite the partition function
as Z = tr(e−β H̃ ), where H̃ now acts on the virtual indices
as opposed to the physical indices. Hence, we call H̃ the
effective entanglement Hamiltonian. Equally important, the
partition function can be understood as that of a classical
model (in one higher dimension) if the Boltzmann weights
of all configurations are nonnegative. The above holds for
general extensive partitions, yet it is particularly interesting
at the symmetric partition, where the partition function of the
critical quantum model can be studied using Monte Carlo or
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PHYSICAL REVIEW B 90, 085137 (2014)
Up to a constant λ, each transfer matrix is nearly the
identity. Therefore we can write T N λ exp[ξ2N λ−1 U2L ⊗
U2R ]. The Suzuki-Trotter expansion (eA eB ≈ eA+B for small
A,B) allows us to ignore the commutator arising from the
overlap of T N1 and T N2 (i.e., the green and blue boxes in
Fig. 3) and write
{
N1
{
N2
i
Z ≈ tr(e−nH̃ ),
N
H̃ ≡ −
ξ2 1 λ−1 U2L i ⊗ U2R i+1
i+1
odd i
FIG. 3. (Color online) The partition function corresponding to
the entanglement Hamiltonian from an extensive partition, tracing
out blocks of N1 sites and leaving N2 sites in between. Boxed in blue
and green are the building blocks of the partition function, namely,
the transfer matrices propagating two virtual degrees of freedom from
one layer to the next.
tensor renormalization-group methods to attain approximate
critical exponents.
Choosing the two subsystems A and B of the extensive
partition to be alternating blocks of N1 and N2 spins,
respectively, we follow the general procedure outlined above
and obtain the tensor network state (TNS) structure of the
partition function, shown in Fig. 3. Interestingly, this TNS can
be viewed as a partition function for the virtual degrees of
freedom (depicted by and hereafter referred to as dots) located
at the ends of each block. Each “ladder diagram” obtained
by contracting physical indices (blue or green boxes in Fig. 3)
plays the role of a time evolution operator acting on two virtual
sites. The value of the ladder can be evaluated by defining the
transfer matrix
T acting on the auxiliary indices, defined as
σ
(M σ ∗ )βδ (see Fig. 1). The two basic building
Tαβ;γ δ = σ Mαγ
blocks boxed in Fig. 3 are then given by T N1 ,T N2 ∗ . When
we permute the vertices and view [T N1 ]αβ;γ δ as a mapping
from auxiliary indices αγ to βδ, it describes the “imaginary
time evolution” of the virtual sites i,i + 1 for odd i and on
odd-numbered rows. Similarly, T N2 acts on sites j,j + 1 for
even j and even-numbered rows.
Further simplification can be made by considering large
N1 ,N2 ; in this limit, the transfer matrix T N1(2) will be dominated
by leading eigenvalues of T . As is well known for MPSs, by
transformations in the virtual indices one can always transform
T into a canonical form. As we show in the Supplemental
Material [28], for the canonical T in the limit of large N1 ,N2 ,
neighboring dots become almost decoupled, and T N has the
form of
T N ≈ λ(1 ⊗ 1) + ξ2N U2L ⊗ U2R ,
(4)
where λ > 0 is a constant, 1 is the identity operator acting
on a single dot, ξ2 is the second largest eigenvalue of the
double tensor of the MPS, and U2L,R are the corresponding
left and right eigenvectors, respectively (they are, nonetheless,
operators acting on a single dot). The tensor product above is
that between two adjacent dots. The key point is that MPSs
which represent gapped ground states have a nondegenerate
largest eigenvalue of the double tensor. This allows us to
analyze the large N, “weak-coupling” limit.
−
ξ2N2 λ−1 U2L j ⊗ U2R j +1 ,
(5)
(6)
even j
up to a constant.
IV. APPLICATION TO HALDANE PHASE
We now demonstrate this procedure explicitly for the
BES of the Haldane phase of the spin-1 chain [29]. This
is a topologically nontrivial phase protected by either time
reversal, a dihedral subgroup of rotations (detailed later),
or inversion symmetry [19,30]. We begin by analyzing a
representative of the Haldane phase: the Affleck-KennedyLieb-Tasaki (AKLT) matrix product state [31],
2 +
2 −
1 z
+
0
−
σ , M = −√ σ , M = −
σ . (7)
M =
3
3
3
Here, ±,0 stand for S z = ±1,0, and σ are the Pauli spin
matrices, with σ ± = 12 (σ x ± iσ y ). The transfer matrix is
i∗
i
σβδ
= 23 (δαβ δγ δ − 12 δαγ δβδ ). Viewing
Tαβ,γ δ = 13 i=x,y,z σαγ
T as a two-site operator acting on the auxiliary indices, we
can write
(8)
T = 12 1 ⊗ 1 − 13 σ · σ ∗ .
This transfer matrix corresponds to ξ2 = − 13 , U2L = σ i , U2R =
σ i∗ in the generic formula, with a threefold degeneracy in the
eigenstates. After a unitary transformation on the odd sites
σ ∗ = −σy σ σy , the effective entanglement Hamiltonian has
the Heisenberg form:
1 N1 1 N2 H̃ ≡ − −
Pi,i+1 − −
Pj,j +1 ,
3
3
even j
odd i
up to a constant. Here P is the projection operator of two
spin-1/2s onto the singlet state.
Hence we find that for large even N1 ,N2 , the entanglement
Hamiltonian has the same spectrum as the antiferromagnetic
spin-1/2 Heisenberg chain with alternating nearest-neighbor
couplings. A quantum phase transition occurs at the symmetric
partition N1 = N2 , where the BES describes a spin-1/2
translation-invariant Heisenberg chain. The quantum phase
transition can also be understood in terms of the original
spin-1 model. In the limit N1 N2 , region A consists of
isolated blocks, while in the opposite limit, N1 N2 , it is
in the Haldane phase. Therefore the phase transition is one
between the Haldane phase and a trivial product state, driven
by translation symmetry breaking. Indeed, this transition is
known to be described by SU (2) level-1 Wess-Zumino-Witten
085137-3
TIMOTHY H. HSIEH, LIANG FU, AND XIAO-LIANG QI
k
l
i
j
PHYSICAL REVIEW B 90, 085137 (2014)
the full SO(3) symmetry group is not necessary to protect
the topological Haldane phase. Hence we now consider
MPS ground states which are slightly perturbed away from
the AKLT state, and we analyze the nature of the bulk
entanglement Hamiltonian in such cases. For this purpose, it is
useful to make the spin symmetries manifest by parameterizing
these MPSs as [36]
M x = aσ x ,
FIG. 4. The partition function from the entanglement Hamiltonian at the symmetric partition is identified as a classical partition
function. (top) The transfer matrix is interpreted as a vertex interaction
between Ising variables (large dots). (bottom) The resulting classical
model is defined on a new lattice.
(WZW) theory [32–34], the same conformal field theory that
describes the Heisenberg spin-1/2 chain.
Although the above derivation applies generally to any
extensive partition of any MPS representation of a gapped
ground state, it is valid only in the large N1 ,N2 limit, an
approximation which may not hold in general in higher
dimensions. Hence we now provide a complementary analysis
that (1) holds for arbitrary N1 ,N2 and (2) generalizes readily to
higher dimensions: we view the partition function as that of a
(two-dimensional) classical model. Because we are interested
in the topological phase transition expected from BES, we
focus on the symmetric partition N1 = N2 = N for the AKLT
state. Upon redrawing the tensor network so that each set of
N contracted physical indices serves as an interaction vertex
between four spin-1/2 nodes (Fig. 4), we find that the classical
partition function is that of a particular six-vertex model on a
square lattice with vertex weights
ij
j
Vkl = δki δl + λδji δlk ,
λ=
(−3)N − 1
.
2
In this model, there are two possible states on each link of
the square lattice, and the weight of each configuration in
the partition function is given by the product of the above
vertex terms. It is remarkable that the AKLT wave function
“contains” such six-vertex models, which are exposed by BES.
For the above parameters, the model is critical, equivalent to
the four-state Potts model, and is described in the continuum
limit by the level-1 SU (2) WZW theory [35].
V. GENERALIZATIONS
This particular universality class is a consequence of the
SO(3) symmetry of the AKLT MPS. However, recall that
M y = bσ y , M z = cσ z ,
(9)
where a,b,c are real numbers. Comparing (9) to (7), we are
simply using a new basis: |± = √12 (x ± iy),|0 = z. When
a = b = c, the MPS is the SO(3) symmetric AKLT state up to
an overall normalization, and when two of the coefficients
are equal, the MPS has at least a U (1) symmetry (in the
plane corresponding to those two coefficients). Finally, when
all three parameters are different, the MPS still has dihedral
symmetry generated by rotations by π about the x,y,z axes.
Parameterizing small perturbations to the AKLT state by
a = 1,b = 1 + δ,c = 1 + , we proceed as above and find that
the entanglement Hamiltonian from the symmetric partition in
the large N and small δ, limit is given by the XY Z spin-1/2
chain (see the Supplemental Material [28]). More specifically,
the spin symmetry of the MPS ground state is in one-toone correspondence with the symmetry of the entanglement
Hamiltonian: the AKLT MPS yields the Heisenberg chain,
the MPS with U (1) symmetry yields the XXZ chain, and
the most general MPS of the form (9) yields the XY Z chain.
Since the XY Z chain either is critical or spontaneously orders
along a direction [37–39], the corresponding entanglement
Hamiltonian either is critical or has ground-state degeneracy,
both of which are consistent with our general arguments [23].
Because of the two free parameters in the MPS, the
resulting entanglement Hamiltonian will generically be in the
gapped part (with ground-state degeneracy) of the XY Z phase
diagram. If the MPS is fine-tuned so that the corresponding
entanglement Hamiltonian lies on one of the critical lines,
then the universality class can be simply extracted from the
XY Z model. All critical lines in the XY Z phase diagram
map to the XXZ chain [39], which in turn can be mapped
via bosonization to (critical) Luttinger liquids with the
Luttinger parameter depending on the XXZ anisotropy. We
conclude that such critical theories describe the transition
of the Haldane phase to a topologically trivial dimerized
phase.
Alternatively, for finite N , we can also map the partition
function of the entanglement Hamiltonian for this wider class
of MPSs to classical models in two dimensions, as we derived
a six-vertex model from the AKLT state. We find that the
general class of MPSs of the form (9) maps into eight-vertex
models [37], although in some cases the weights of some
configurations may be negative.
VI. SUMMARY AND OUTLOOK
By using tensor networks to construct the partition function
of the critical bulk entanglement Hamiltonian, we gain much
insight into the topological phase transition revealed by BES.
Previously, topological phase transitions have been described
by the condensation of fractionalized excitations or the delocalization of edge states [40]; in both cases, the original degrees
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PHYSICAL REVIEW B 90, 085137 (2014)
of freedom of the model are overshadowed by emergent
ones. Thanks to the partition function and the tensor network
framework, we have a clear picture of the virtual degrees of
freedom interacting and giving rise to the phase transition. As a
by-product of this procedure, we attain classical lattice models
of quantum critical points; as an example, we derived critical
six-vertex models from the AKLT wave function. We note that
this tensor network implementation of BES generalizes readily
to higher dimensions, in which interesting quantum-classical
mappings may await.
Note added. Recently, we noticed numerical results [41]
for a fixed extensive partition of the AKLT chain in which
alternating blocks of two sites are traced out. We also noticed
a recently posted study relating entanglement properties from
extensive partitions of spin chains to classical models [42].
These results support our conclusions.
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ACKNOWLEDGMENTS
We thank T. Senthil for informing us of the relation
between the Heisenberg model and the four-state Potts model.
T.H. is supported by NSF Graduate Research Fellowship No.
0645960. L.F. is supported by the DOE Office of Basic Energy
Sciences, Division of Materials Sciences and Engineering,
under Award No. DE-SC0010526. X.-L.Q. is supported by
the National Science Foundation through the Grant No.
DMR-1151786.
085137-5